Self-adjoint extensions and unitary operators on the boundary

Abstract

We establish a bijection between the self-adjoint extensions of the Laplace operator on a bounded regular domain and the unitary operators on the boundary. Each unitary encodes a specific relation between the boundary value of the function and its normal derivative. This bijection sets up a characterization of all physically admissible dynamics of a nonrelativistic quantum particle confined in a cavity. More- over, this correspondence is discussed also at the level of quadratic forms. Finally, the connection between this parametrization of the extensions and the classical one, in terms of boundary self-adjoint operators on closed subspaces, is shown.

16 pages

References in corpus (4)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Quantum systems with time-dependent boundaries
• Quantum Physics and Fluctuating Topologies: Survey
• Boundary Effects in Bosonic and Fermionic Field Theories

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